No, not unless you SEVERELY over load it, which I doubt you could. While I know of one case where one blew up on somebody, that was from putting several thousands of volts into one that shouldn't have had more than about 300 impressed on it. I was a diesel mechanic in a previous work life and did quite well on the electrical side of things, especially on exams. What you'd be doing is checking the potential between the pc case and some ground, which isn't really any different than if you put across a battery (DC) or across your household electrical outlet. The meter won't conduct current unless you've got it set to measure current, so don't do that if it's a multi-meter. Otherwise, as I said, you're just measuring potential. If you get 50V, that's how much is leaking across that wire. NOTE 30VAC is enough to kill you, so I suggest you don't screw around with this issue. If you've got leakage to ground from a power supply or something on that pc case you are playing with danger. Usually these things don't get better - they get worse.
Do I have to point out that you don't hold the metal prongs of the meter leads in your hands?? It should go without saying that you need to be insulated, but assumptions can be dangerous.
Well I'm blowed! Same here! I worked on lorries for several years (for The Americans, lorries is an English term for a lorry? In America that would be a truck! Possibly!)
I was also pretty good at sussing out the electrical faults. Mind you I had the opportunity to look under a HGV bonnet recently and there was this big box under there, I think it said something like danger 7000 volts! It was something to do with the electronic diesel injection system. Made me realise that I probably shouldn't consider tinkering with someone's lorry, not without getting a refresher on health and safety!
If you had a tingle when touching a wire attached to the chassis, you have a flawed tower. Something isn't right inside the box. If the case isn't grounded then you have a potential danger. That wire you mentioned that leads to the radiator will probably provide a decent ground, but the fact that you needed one is not good.
Did you buy that mini-tower or build it? If you bought it, you might wish to talk to the tech service department of the seller and ask if they have been having ground fault problems in that particular model. If you built it, something wasn't connected correctly.